JP4547427B2 - Gas cell using two parabolic concave mirrors and gas sensor manufacturing method using the gas cell - Google Patents

Description

  The present invention relates to a method of manufacturing a gas cell, which is the most important part in a gas concentration measuring apparatus using an NDIR (non-dispersive infrared) system. More particularly, the present invention relates to a method of manufacturing a gas cell with an optical cavity that is suitable for a variety of applications and has a simple geometry that facilitates optical path analysis and uses such a gas cell. The present invention relates to a method for manufacturing a gas sensor.

  There is a growing interest in the atmospheric environment, and many people are interested in preventing unforeseen accidents by accurately detecting harmful gases generated in the working environment as well as harmful gases in the atmosphere. For this reason, there is an increasing need for not only large-scale gas sensors but also portable or portable gas sensors that can be used in a small room, and it is necessary to reduce the size and weight of gas sensors. Therefore, it is very important to reduce the size and weight of the gas cell, which is a core part for detecting gas, and it is necessary to manufacture the most efficient gas cell among the limited sizes. Since NDIR gas cells measure the light absorption of gas through light, attempts have been made to lengthen the light path and many results have been achieved. .

  In order to obtain a longer light path in a confined spatial region, it can be said that a method in which light is reflected many times in the light cavity of the gas cell using a mirror is the most useful method. In addition, optical cavities having various geometric structures have been proposed, but it is quite difficult to lengthen the optical path. This is because light sources and photodetectors actually have such a great influence that they cannot be ignored in the optical cavity, which is a major limitation in the production of optical cavities. Therefore, in order to overcome such restrictions, attempts have been made to overcome such restrictions by arranging mirrors in various geometrical ways, but the geometric structure is rather complicated. It is cumbersome to analyze it. In other words, since an optical cavity with a complicated geometric structure cannot be easily deformed, an optimal optical path is searched for using the results obtained by various simulations in order to deform the optical cavity. I have to put it out. Further, if even a slight variation occurs in the factor for analyzing the optical cavity due to the manufacturing defect of the optical cavity, it is impossible to obtain a desired optical cavity. Therefore, the optical cavity must be manufactured very strictly. Therefore, there is a problem that a great amount of cost and time are required for manufacturing the optical cavity.

  An object of the present invention is to provide an example of an existing gas cell in the manufacture of a gas cell, which is an important part of a gas sensor that measures the concentration of a gas by utilizing the absorption characteristic of the gas to light by the NDIR (non-dispersive infrared) method. The object is to propose a method for manufacturing and analyzing gas cells with much better optical measurement properties.

The gas concentration measurement method using the NDIR method has attracted attention recently because it is inexpensive and has high measurement accuracy and precision. The principle of such a system is to use the absorbance of a specific wavelength with respect to light of gas, and measure the degree of absorption of light when the gas is irradiated with light of that wavelength. The concentration can be calculated. More specifically, for example, in the case of carbon dioxide (CO ₂ ), the best absorption characteristics are shown with infrared rays having a wavelength of 4.3 μm, but in order to measure the concentration of carbon dioxide, the infrared rays of 4.3 μm are irradiated and the concentration of carbon dioxide is 0. In this case, the concentration of carbon dioxide can be calculated from the ratio of the intensity of infrared rays detected from the photodetector and the intensity of the remaining infrared rays after being absorbed by carbon dioxide.

  In such a case, even if the gas has the same concentration, the longer the path (path) through which the light travels through the gas, the smaller the intensity of the light measured from the photodetector, so that the light is incident as a result. Since the ratio of the intensity of the measured light and the intensity of the measured light is increased, more precise measurement is possible. After all, in the NDIR gas concentration measurement method, it can be said that the most important requirement is to manufacture an optical cavity that provides a longer optical path in a limited spatial region.

  Therefore, in order to satisfy these requirements, the present invention provides an optical cavity manufactured so that the path of light can be made very long by using two parabolic concave mirrors sharing a focal point and an optical axis. The method of manufacturing the gas cell used and its analysis method are to be presented. Therefore, the present invention makes it possible to manufacture a gas sensor that can measure the concentration of gas more accurately and accurately by presenting a gas cell including an optical cavity having superior properties than existing gas cells.

  As described above, in order to manufacture a gas cell for obtaining an effective optical path within a limited size, the following conditions must be satisfied. That is, the present invention presents a method for producing a gas cell that satisfies such conditions and for appropriately analyzing such a gas cell.

  1) When designing optical cavities, lenses and mirrors with appropriately shaped geometric structures will be used, but such geometric structures should be as simple as possible. This not only facilitates system analysis, but also saves money and time in manufacturing.

  2) The optical system in the optical cavity should be stable. In other words, for small impacts that may occur during external shocks or gas cell manufacturing processes, even if the optical path deviates slightly from the desired path, the light can be converged stably on the photodetector. There must be. In order to satisfy such conditions, stable output (light detection) must be possible even with variations in input conditions (light source conditions).

  Therefore, the present invention proposes a method for manufacturing an optical cavity that satisfies the above-described conditions and a method for manufacturing a gas cell using an optical cavity having a longer optical path length than an existing optical cavity.

  In the present invention, as shown in FIG. 1, two quadratic function parabolic concave mirrors having different focal lengths are arranged to face each other so as to share the focal point and the optical axis. In addition, a light source is arranged at the point where the two concave mirrors intersect so as to face the focus, and a photodetector is arranged on the optical axis. With such a configuration, the light emitted from the light source passes through the focal point, is reflected several times in the concave mirror, converges to the optical axis, and is eventually detected by the photodetector. Such convergent light can be obtained even when the light source is slightly displaced from the focal point due to manufacturing errors. Therefore, the present invention provides a system that is simple in structure and easy to analyze by using a quadratic function parabolic mirror under the above-described conditions. Such a system has very stable light detection characteristics. The configuration of the present invention will be described in detail below.

In order to achieve such a configuration, a function of a concave mirror having the following characteristics (1) and (2) is obtained, and an optical convergence system using the function is provided.
(1) Incident light that passes through the focal point of the concave mirror is reflected by the mirror surface and then travels parallel to the optical axis.
(2) The light incident parallel to the optical axis is reflected by the mirror surface and then travels through the focal point.

As described above, since the optical cavity of the gas cell proposed in the present invention uses a parabola of a quadratic function, it has a characteristic that it is very easy to manufacture and analyze. The variables to consider when using the present invention to produce an optical cavity are p, p ′, L ₁ , and L ₂ , and appropriately adjusting such variables as presented above. So, a relatively simple mathematical relational expression was also presented as a method for producing and analyzing the optimum optical cavity. By appropriately adjusting such a relational expression so that a gas cell having desired characteristics can be manufactured, an optimal gas sensor can be manufactured.

  In addition, the optical cavity presented in the present invention can stably converge light on the photodetector even if the optical path deviates slightly from the desired path due to external impacts or small mistakes that can occur during the manufacturing process of the gas cell. By adopting a structure, stable output (light detection) is possible even when the input conditions (light source conditions) change. In other words, the optical cavity of the gas cell presented in the present invention can stably converge the light to the photodetector even if there is some variation with respect to the ideal optical path, This has the effect of saving waste.

  In addition, the optical cavity presented in the present invention has a very long optical path as compared with the existing optical cavity even when the above example is seen. Therefore, a more precise and accurate gas cell can be manufactured.

  Most existing optical cavities are searched for optimum optical path conditions by a simulation method by combining various concave mirrors, convex mirrors and lenses. However, in such a case, not only a lot of trial and error is repeated, but a lot of human power is used, and a lot of time and cost are used in the actual manufacturing and test (testing) process. This will increase the price.

  As described above, the present invention is very simple to construct and is not so difficult to mathematically analyze. Therefore, the manufacturing cost of the optical cavity can be greatly reduced. As a result, it is possible to eliminate the cause of excessive price increase and to respond appropriately to market demands.

Hereinafter, the configuration of the present invention will be described in more detail with reference to the drawings.
FIG. 1 is a diagram of an optical cavity using two two-dimensional parabolic concave mirrors according to one embodiment of the present invention.

In these quadratic parabolic concave mirrors, light incident toward the focal point travels parallel to the optical axis after being reflected by the mirror surface, and light incident parallel to the optical axis passes through the focal point after being reflected by the mirror surface. To do. Utilizing the reflection characteristics of the concave mirror of such a quadratic function, the two concave mirrors face each other so as to share the focal point with respect to the two quadratic functions having focal lengths p and p ′ as shown in FIG. To form an optical cavity. Further, a light source is arranged so as to be focused on A ₀ , and a photodetector is arranged on the optical axis (p, 0) in the −x direction.

  The light emitted from the light source toward the common focal point is internally reflected several times due to the characteristics of the parabolic concave mirror, converges to the optical axis, and can finally be detected by the photodetector.

  In such a case, the length of the optical path can be adjusted according to the conditions of p and p ′. For example, when the beam size and the detector diameter are 4 mm and 4 mm respectively under the conditions of p = 15 mm and p ′ = 13.5 mm, the optical path length of about 1026 mm can be obtained by the analysis method of the present invention.

1. Derivation of concave mirror function In order to derive the function of the concave mirror having the above characteristics (1) and (2), a simple differential equation is used.

  FIG. 2 presents a diagram that satisfies the above two conditions for an arbitrary function as a diagram of a quadratic function parabola that forms an optical cavity according to one embodiment of the present invention.

が成立するので、(x、y)は(1)式のような方程式を満足する。

For a mirror that takes the form of an arbitrary function y = f (x) (201), light (202) incident parallel to the x-axis is reflected on the mirror's A (x, y) (203) and the origin (210) (204), the normal (205) from the reflection principle that the incident angle α (207) and the reflection angle β (206) are the same as the equation of the normal (205) passing through A (203) Passes through point B (208) on the x-axis. At that time, the condition of α (207) = β (206) = γ (209) is satisfied,

(X, y) satisfies the equation (1).

When equation (1) is converted as two-dimensional polar coordinates (γ, θ), equation (2) is obtained.

By rearranging equation (2), equation (3) is derived.

When substituting cosθ-1 = z to find the solution of the differential equation of equation (3), dz = −sinθdθ, so equation (3) can be expressed as the following equation (4).

In FIG. 2, when y = 0, if x = −p ₀ (p ₀ > 0), then γ = p ₀ and θ = π, so that C ₀ = −2p ₀ can be obtained from equation (4). Substituting this into equation (4) and rearranging results in equation (5).

If the equation (5) is rearranged, the curve of the secondary parabola with respect to y as in the equation (6) can be derived.

A concave mirror manufactured with such a second-order parabolic curve derived from FIG. 2 and equation (1) has the following 'reflection characteristics'.
-Light incident parallel to the optical axis (x-axis in FIG. 2) is reflected by the concave mirror of the second-order parabolic curve and passes through the focal point (the origin (O) in FIG. 2).
-Light incident through the focal point is reflected by a concave mirror with a quadratic parabolic curve and travels parallel to the optical axis.

2. Characteristics of optical cavity configured as concave mirrors of two quadratic function parabolas The characteristics of the optical cavity will be described below with reference to FIG. Here, the optical cavity uses two parabolic concave mirrors that are arranged so as to face each other and share the focal point and the optical axis, but have different focal lengths.

FIG. 3 is a diagram showing the path characteristics of an optical cavity according to an embodiment of the present invention, and y ² = −4p (xp) (301) and y ² = 4p ′ (x + p ′) (302). 2 illustrates an optical cavity realized by combining two concave mirrors, each corresponding to a quadratic parabola. Each vertex is (p, 0) (305) and (-p ', 0) (306), where (0 <p'<p). Each focal point F is the origin (303) and the optical axis is the x-axis (320), which are shared.

とする。光源から放出された光は閉鎖された光空洞で１回循環(A₀ 304 → B₀ 307 → C₀ 308 → D₀ 309)を経て、A₁に到達する。その後、光は類似のルート（A₂ → A₃ →A₄ ... A_n → ... A_∞）を進む。前述したようにA_∞は(p, 0)であり、この点に検出器を位置させる。このようなA_N座標を求めることで、光空洞内部での光の進行特性を特定することができる。そのため、A_N、B_N、C_N、及びD_Nを（７）式のように定義する。

The position of the light source (304) may be set to any position as long as the emitted light passes through the focal point and is reflected by the concave mirror (301). In the present invention, for convenience of explanation, the light source is arranged at A ₀ which is one of the two intersections of the concave mirror (301) and the concave mirror (302) and is located in the + y direction. However, the position of such a light source (304) is meant only as an example, and it is clear that the gist of the present invention is not limited to such a position of the light source. The coordinates of the point where the light source (304) is located are

And The light emitted from the light source reaches the A ₁ through a closed optical cavity through one circulation (A ₀ 304 → B ₀ 307 → C ₀ 308 → D ₀ 309). After that, the light follows a similar route (A ₂ → A ₃ → A ₄ ... A _n → ... A _∞ ). As described above, _A∞ is (p, 0), and the detector is positioned at this point. By determining such an A- _N coordinate, it is possible to specify the light traveling characteristics inside the optical cavity. Therefore, A _N , B _N , C _N , and D _N are defined as in equation (7).

Here, A ₁ = (α ₁ , β ₁ ) is expressed as α ₀ and β ₀ , and this is generalized and A _n = (α _n , β _n ) is expressed as α ₀ and β ₀ , so that the light travels Analyze the road. For this reason, B ₀ , C ₀ , and D ₀ are first expressed as α ₀ and β ₀ , respectively.

In FIG. 3, A ₀ and B ₀ are points on y ² = −4p (xp) and y ² = 4p ′ (x + p ′), respectively, and are on a straight line passing through the origin. The relationships as shown in equations 8), (9), and (10) are established.

(8), (9), (10) of alpha _'0 and beta' and rearranging the binary quadratic equations for ₀ for alpha _'0 (11) expression is induced, when seeking this solution It becomes like (12) Formula.

Here, since α ₀ and α ′ ₀ are symmetrical points with respect to the origin, the signs are opposite to each other. Therefore, in equation (12), only α ′ ₀ = − (p ′ / p) α ₀ is satisfied. Β ′ ₀ is derived from the equation (10). (Refer to equation (13))

はx軸に平行な直線であるため、β’’₀=β’₀が成立し、また、C₀はy² = -4p(x-p)上の点であるため、C₀は(14)式のように求められる。

Depending on the characteristics of the function of the secondary parabola

Is a straight line parallel to the x-axis, so β '' ₀ = β ' ₀ is satisfied, and C ₀ is a point on y ² = -4p (xp), so C ₀ is the formula (14) It is required as follows.

Here, if A ₀ is an arbitrary point on y ² = −4p (xp), the route A ₀ → C _{0 is} eventually symmetric with the route C ₀ → A ₁ . That is, A ₁ can be displayed as the coordinates of C ₀ and can be displayed as the coordinates of A ₀ . So, the coordinates A ₁ is displayed as (15). (Here T ≡-(p '/ p) (<0).)

From the result of equation (15), it can be considered that the relational expression of _An and _An-1 is the same as that of equation (16).

_Similarly , from the equations (13), (14), and (16), B _n , C _n , and D _n are derived as the following equation (17).

As can be seen from equation (16), it can be seen that (α _n , β _n ) converges to (p, 0) as n increases. That is, it can be seen that the light emitted from the light source (304) reciprocates between (p, 0) and (−p ′, 0) and finally converges on the x-axis (320) that is the optical axis. . When the photodetector (321) is placed at the point (p, 0) on the x-axis (320), which is the optical axis, so as to face the origin (focal point), the light emitted from the light source (304) and passing through the focal point is All are converged on the photodetector (321).

3. Length of the optical path in the optical cavity _Let the length of the optical path of A _n-1 → A _n in the optical cavity be the circulation length L (A _n ) and find L (A _n ). The purpose of obtaining L (A _n ) is to calculate the total length of the light path until the light emitted from the light source reaches the photodetector. That is, the total length of the optical path is obtained by repeatedly obtaining the length of the path during which light circulates once in the optical cavity and adding it up. The total length of the light path is a very important factor that affects the performance of a gas sensor, and if you know how to analyze it, the most efficient gas sensor can be manufactured at a low cost when manufacturing gas sensors by application. It becomes possible.

であるため、光の循環長さをそれぞれ求め、nに関して合算する。(16)式と(17)式から(18)式が導出される。

The total length of the light path is

Therefore, the light circulation lengths are obtained and summed up for n. Equation (18) is derived from Equation (16) and Equation (17).

Using formula (18) and formula (8), (9), (16), and (17), the length L (A _n ) after one cycle is calculated using (19) ) Is derived.

From the equation (19), the total length L of the optical path when the light emitted from the light source located at (α ₀ , β ₀ ) circulates n times in the optical cavity can be expressed as the equation (20).

Equation (20) is constructed as two functions related to T. Assuming that L = L (T) = F ₁ (T) + F ₂ (T), F ₁ (T) and F ₂ (T) are expressed as equations (21) and (22).

The reason for dividing the total length L of the optical path into two functions is to calculate the contribution of the two functions to the total length of the optical path. It can be seen that as N (the number of circulations) increases, F ₁ (T) increases proportionally, but F ₂ (T) decreases. Therefore, when N is sufficiently large, the total optical path length L is considered to be mainly affected by F ₁ (T). The relative contribution of F ₁ (T) and F ₂ (T) is expressed as G (T) in equation (23).

The value of α ₀ = p−p ′ presented in FIG. 3 is substituted into equation (23). From FIG. 3, it is understood that the circulation length increases when the light source is arranged at the outermost side of the optical cavity. As will be described later, such a point A ₀ is an optimum position for arranging the light source. However, the position of the light source of the present invention is not limited to A ₀ of Fig. 3 is of course.

Substituting α ₀ = p−p ′ into equation (23) and rearranging results in equation (24).

In Equation (24), T takes a negative value and N takes a positive value, so the condition as in Equation (25) is satisfied. Since T ^4N is a positive number, G (T) satisfies the following condition in a section where -1 <T <0.

From equation (25), L when the number of circulations N is sufficiently large can be approximated as equation (26).

4. Calculation of the number of circulations N based on the conditions of p and p ′ based on the beam size and the size of the photodetector In the present invention, the light traveling in the optical cavity is expressed by the following equation (16): It circulates in the optical cavity and converges on the optical axis. As can be seen from equation (16), the speed of convergence can be adjusted according to the condition of T. That is, the convergence speed decreases as T is closer to -1, and when T is -1, an infinite loop is obtained. The convergence speed increases as T is closer to 0.

In practical use, since the photodetector has a certain size, light traveling in the optical cavity is detected by the photodetector after circulating through the optical cavity a limited number of times. Therefore, the number of circulations N can be adjusted by adjusting the T value. However, since the light sources have a certain size, after light emitted from the light source is circulated once, when it reaches the A _1, the light may interfere with the light source. In consideration of such conditions, the number of circulations until the light traveling through the optical cavity converges on the photodetector must be calculated.

FIG. 4 illustrates the interference (circle 401) by the light source and the condition (circle 402) under which light is detected depending on the size of the photodetector. A circle (403) is an enlarged view of the circle (402). Since light travels in bundles, not lines, it has a constant beam size. Thus, since the light has a predetermined beam diameter (L _{1 in} FIG. 4), it must not overlap with the light source when it reaches A ₁ after it has circulated once. In other words, such duplication means loss of light and will immediately cause the performance of the gas sensor to deteriorate.

  Since the size of the light emitted from the light source (beam diameter) is the same as the size of the light emission port of the light source, it is assumed that the size of the light source and the beam diameter are the same.

Therefore, the condition that the light source and the light shown in a circle (403) in FIG. 4 do not overlap is given by the equation (27).

When the light source is located as shown in FIG. 3 described above, β ₀ , β _1, and sin θ in Expression (27) are expressed as Expression (28).

Substituting Eq. (28) into (27) and rearranging results in (29), and Eq. (29) ensures that the light emitted from the light source does not overlap with the light source after circulating once. It is a condition for. That is, when manufacturing the optical cavity according to the present invention, the p and p ′ values must be adjusted so as to satisfy Equation (29).

Further, as shown in a circle (404) which is an enlarged view of the circle (402), if the light from the light source circulates through the optical cavity N times and then converges on the photodetector, the Nth circulation of light is performed. In order to be effective, when it is considered that light is detected when light with half the beam diameter overlaps with the photodetector, the light detection condition considering the size of the photodetector is expressed by Equation (30). Must be met. Here, it is assumed that light is detected by the photodetector when half of the beam diameter and half of the cross section of the photodetector overlap.

であるため、(30)式は(31)式として表現される。

Similarly, substituting the conditions of FIG.

Therefore, Expression (30) is expressed as Expression (31).

If both sides of equation (19) are arranged by taking the natural logarithm, equation (32) is derived.

Equation (32) is a condition for the maximum number of cycles for p, p ′, L ₁ , and L ₂ .
Therefore, when the gas cell of the present invention is manufactured, the conditions of p and p ′ are applied from the equation (29), and the number of light circulations is calculated from the equation (32) accordingly. For example, when manufacturing an optical cavity under the conditions of L ₁ = 4mm, L ₂ = 4mm, p = 10mm, p '= 9mm, this condition satisfies the equation (29) and the number of circulations N from the equation (32) = 7 is required. Substituting this into the equation (26), the total length of the optical path in the optical cavity is calculated to be about 532 mm.

5. Stability analysis against optical path deviation When measuring the concentration of gas in order to produce an efficient gas cell, the intensity of light emitted from the light source is preferably as large as possible. However, it is difficult to use light having a desired intensity in consideration of the physical properties of the luminescent material of the light source and the lifetime of the light source. Therefore, in order to use the light of limited intensity most efficiently, the light emitted from the light source using a convex lens or a concave mirror is collected to collect the isotropically emitted light in one direction. There is a method to collect in the common focus of the cavity. When a convex lens or a concave mirror is used, light can be collected in a single focal point in theory, but it is actually very difficult to actually pass light through one point. In addition, if the focus is on manufacturing in this way, the cost and time may be required to manufacture the gas cell. Even if many efforts were made to allow light to pass through a single point, the optical system inside the gas cell was misaligned due to external impacts or small mistakes in the manufacturing process. Sometimes light travels out of focus. In such a case, since the amount of light reaching the photodetector is reduced, the measurement efficiency is lowered, and eventually the performance is lowered.

  The optical cavity of the gas cell proposed in the present invention provides very good stability over the above case. This will be described with reference to FIGS. 5 and 6.

  FIG. 5 shows the procedure for analyzing the optical path when light travels out of the focus of an optical cavity with two parabolas of the same focal length. That is, FIG. 5 is a diagram for explaining the stability in the case where the parabolic mirrors (501) and (502) of the two quadratic functions constituting the optical cavity have the same focal length (p = p ′). is there.

  Even when the focal lengths of the two quadratic function parabolas are different from each other (illustrated in FIG. 6), it is possible to analyze the light path based on the procedure of FIG.

In FIG. 5, when the focal length is the same, the light emitted from the light source (503) and passed through the common focal point returns to the original position again. That is, the light emitted from the light source (503) at the point A accurately toward the focal point returns to A again through B, C, and D. This can be confirmed by the equation (16) (where T = -1). Reaches the B ₀ 'is not a B ₀ progressed slightly off the light focus emitted from the light source (503), each degree of deviation of the light in the x-axis and y-axis directions epsilon _{(0) 1} and If δ _{(0) 1} and the light has a path of A ₀ (503) → B ₀ ′ (506) → C ₀ ′ (508) → D ₀ ′ (510) → A ₁ ′ (504), then A ₀ through light is once circulated release can determine coordinates of 'a ₁ when it reaches the' a ₁ from, it is possible to generalize the 'a ₁ from the relation between' a ₀ and a ₁ As a result, a light path is obtained.

The coordinates of A ₀ (503), B ₀ ′ (506), C ₀ ′ (508), D ₀ ′ (510), and A ₁ ′ (504) are given by the equation (33). Where ε _{(0) 1} , ε _{(0) 2} , δ _{(0) 1} , δ _{(0) 2} , μ _{(0) 1} , μ _{(0) 2} , ν _{(0) 1} , and ν ₍₀₎ Assume that the absolute value of ₂ is sufficiently smaller than the value of p and that their products converge to 0.

First, in order to utilize the symmetry of the optical cavity of the present invention, the case where the light path is A ₀ ′ → B ₀ ′ → C ₀ ′ will be analyzed first. In such a case, for a given ε _{(0) 1} and δ _{(0) 1} , the coordinates of C ₀ ′ may be expressed as α ₀ , β ₀ , ε _{(0) 1} and δ _{(0) 1.} is there. Therefore, since the light is reflected from B ₀ ′ and reaches C ₀ ′, the light reflection law is used.

の勾配をtanθ_AB’とし、B’の法線の勾配を

とすると、三角関数の引き算の公式によりtanΔは次のように表示される。

here,

The gradient of tanθ _{AB '} and the gradient of B' normal

Then, tanΔ is expressed as follows by the trigonometric subtraction formula.

が得られる。(α’₀,β’₀)はy² = 4p(x+p)上の点であるため、この点での法線の勾配である

は(37)式で表される。

From equation (33)

Is obtained. (α ' ₀ , β' ₀ ) is the point on y ² = 4p (x + p), so it is the normal gradient at this point

Is expressed by equation (37).

Therefore, when formulas (36) and (37) are substituted into formula (35) and rearranged, tanΔ is obtained as follows.

の勾配であるため、(33)式の勾配の公式を利用して以下の(39)式を導出できる。

tanΔ is

Therefore, the following equation (39) can be derived using the gradient formula of equation (33).

By substituting the equation (33) into the equation (39) and rearranging, the relationship between ε _{(0) 1} and μ _{(0) 1} can be obtained as expressed by the following equation (40).

Here, the relationship between δ _{(0) 1} and ν _{(0) 1} is not considered. This is because the relationship between δ _{(0) 1} and ε _{(0) 1} and ν _{(0) 1} and μ _{(0) 1} can be derived from the quadratic parabolic equation.

Since the light reaching C ₀ ′ can be regarded as a new light source due to the symmetry of the optical cavity of the present invention, the following equation (41) is derived from equations (33) and (40).

の勾配を

とし、B₀'での法線の勾配を

とすると三角関数の引き算の公式から次の(45)式が誘導される。

Therefore, the coordinates of A ₁ ′ can be obtained from the relationship between ε _{(0) 1} and ε _{(0) 2} . Also, using the light reflection law

The slope of

And the slope of the normal at B ₀ '

Then, the following equation (45) is derived from the trigonometric subtraction formula.

と(38)式と(33)式を利用して(45)式を整理すると、ε₍₀₎₁とε₍₀₎₂の間に次のような関係を導出できる。
ε₍₀₎₁ = -ε₍₀₎₂−−−−−(46) Derived from the normal formula of the quadratic parabolic curve

Using (38) and (33) and rearranging (45), the following relationship can be derived between ε _{(0) 1} and ε _{(0) 2} .
ε _{(0) 1} = -ε _{(0) 2} ------ (46)

By substituting equation (46) into equation (41), the x coordinate of A ₁ ′ can be derived as represented by the following equation (47).

(47) is expressed as ε _{(0) 2} × ε _{(0) 1} = 0, ε _{(0) 2} × μ _{(0) 1} = 0, and 1 / (1-x) = 1 + x (| x | It can be obtained using the condition of << 1) and (46).

As can be seen from equation (47), in a cavity configured as two secondary parabolas having the same p-value and sharing a focal point, the light emitted from the light source (A ₀ ) is emitted from the focal point. Even if it is slightly off (ε _{(0) 1} << p), it turns out that it returns to the original place (A ₁ = A ₀ ) after circulating once in the cavity. Therefore, in the optical cavity of the present invention having different focal lengths (p and p ′ (0 <p ′ <p)) and sharing the optical axis with the focal point, the light emitted from the light source is slightly out of focus. It is possible to derive an assumption that even if the vehicle travels, it does not deviate significantly from the original route (a route that travels accurately through the focal point).

  Based on the above assumption, the light emitted from the light source in the optical cavity of the present invention configured to face each other so that they have different focal lengths and share the focal point and the optical axis does not pass through the focal point accurately. Will be described below and to what extent the light may be out of focus.

FIG. 6 shows the light path when the focal lengths of the two quadratic function parabolic mirrors are different from each other. The two quadratic parabolic mirrors correspond to y ² = −4p (xp) (601) and y ² = 4p ′ (x + p ′) (602) as described above (where 0 <p ′ <p, T ≡-(p '/ p)).

The photodetector (603) is located at the point (p, 0) on the optical axis, and the light emitted from the arbitrary light source A ₀ (604) toward the focal point circulates in the optical cavity as described above. Then, the light is converged on the photodetector and detected. As described above, regularity is searched for in the process in which the light emitted from the light source circulates once, and is generalized for the entire circulation.

Light emitted from an arbitrary light source (604) toward the focal point circulates once through A ₀ (604) → B ₀ (605) → C ₀ (607) → D ₀ (609) → A ₁ (611) The light slightly out of focus circulates once through A ₀ (604) → B ₀ ′ (606) → C ₀ ′ (608) → D ₀ ′ (610) → A ₁ ′ (612). Assuming that the respective coordinates can be expressed by the following equations (48) and (49) with reference to equations (18) and (33).

The light emitted from the light source is slightly out of focus and can deviate by ε _{(0) 1 in} the x coordinate direction and δ _{(0) 1} in the y coordinate direction at the point B ₀ . In this case, as described above, ε _{(0) 1} , ε _{(0) 2} , δ _{(0) 1} , δ _{(0) 2} , μ _{(0) 1} , μ _{(0) 2} , ν _{(0) 1} , and The absolute value of ν _{(0) 2} is much smaller than p, and their product converges to zero. That is, the light emitted from the light source is slightly out of focus based on the fact that, under the condition of p = p ′, the emitted light circulates once in the cavity and then returns to the original light source position. However, suppose that it proceeds slightly off the original route.

の勾配を

とし、

の勾配を

とすると、光の反射法則と三角関数の引き算の公式を適用しつつ、(34)式と(35)式が用いられる。

Therefore, the light path when p = p ′ is not analyzed, as in the case where the light path is obtained under the condition of p = p ′ described above. If light emitted from the light source A ₀ (604) proceeds out of the focus slightly 'is reflected by the C _0' B ₀ to reach,

The slope of

age,

The slope of

Then, equations (34) and (35) are used while applying the light reflection law and the trigonometric subtraction formula.

Therefore, equation (50) is obtained from equation (49).

は(51)式で表される。

Also, B ₀ 'is a point on y ² = -4p (xp), so it is the normal gradient on this point

Is expressed by equation (51).

By substituting the equations (50) and (51) into the equation (35), tanΔ can be expressed by the equation (52).

の勾配であるため、勾配の公式から次の(53)式が成立する。

TanΔ is

Therefore, the following equation (53) is established from the gradient formula.

を得る。また、C₀'の座標をy² = 4p’(x+p’)に代入すれば、

を得る。

Since the equations (52) and (53) are the same, the relationship between ε _{(0) 1} and μ _{(0) 1} can be calculated.
If we substitute the coordinates of B ₀ 'into y ² = -4p (xp),

Get. And if you substitute the coordinates of C ₀ 'into y ² = 4p' (x + p '),

Get.

Since the light reaching C ₀ ′ is reflected and this reflected light can be regarded as a new light source, the relationship between ε _{(0) 2} and μ _{(0) 2} is the following (55 ) Expression.

の勾配を

とし、C₀'での法線の勾配を

とすると、三角関数の引き算の公式から(45)式が再び用いられる。

Since μ _{(0) 2} can be obtained from the relationship between ε _{(0) 1} and ε _{(0) 2} , the coordinates of A ₁ ′ (612) can be obtained after all. Therefore,

The slope of

And the slope of the normal at C ₀ '

Then, equation (45) is used again from the trigonometric subtraction formula.

であるため、(49)式と(52)式を代入して(45)式を整理すると、ε₍₀₎₁とε₍₀₎₂の関係式は次のように求められる。

The slope of the normal at B ₀ 'is

Therefore, by substituting the equations (49) and (52) and rearranging the equation (45), the relational expression between ε _{(0) 1} and ε _{(0) 2} can be obtained as follows.

When the formulas (54), (55), and (56) are applied to the formula (49), the formula (49) can be rearranged with respect to α ′ ₁ as the following formula (57).

(57) from the equation, if the light emitted from the light source (604) proceeds out slightly from the focal point, after circulating once to reach the alpha _'1. In order to generalize this, hereinafter, the prime code is removed from α ′ ₁ . The prime code is used to distinguish light out of focus from light that has passed through the focus, and omitting the prime code for generalization does not limit the present invention. .

The generalized equation (57) is the following equation (58).

Further, since the light reflected from C ₀ ′ can be regarded as a new light source, ε _{(1) 1} and ε _{(0) 2} have a relationship as shown in the following equation (59).

Therefore, the relationship between ε _{(1) 1} and ε _{(0) 2} as shown in equation (60) can be derived from equations (56) and (59).

Here, the product of ε _{(0) 1} , ε _{(0) 2} , μ _{(0) 1} , and μ _{(0) 2} converges to zero. Substituting (56) into (60) and generalizing it yields the following (61).

By substituting equation (61) into equation (58) and generalizing, equation (62) is derived.

In the case of n → ∞, α _n = α _n−1 , and therefore, when α _∞ −p = σ, σ is a negative number having a very small absolute value. (Expression (62) is a convergence function.) Therefore, when n → ∞, expression (63) can be expressed by the following expression (64).

When formula (64) is arranged with respect to σ, formula (65) is obtained.

In the case of n → ∞, if β _n converges to γ, the following relational expression (66) is derived for the + x axis (p) based on the y ² = −4p (xp) expression.

Further, when formula (66) is arranged with respect to γ, formula (67) is obtained. (Here, for convenience, the value of γ is set to a positive number under the condition of ε _{(0) 1} > 0.)

γ is the y coordinate of the point where light emitted from the light source (604) and traveling out of focus finally converges. As shown in FIG. 4, when the diameter of the light detector and L _2, In this case, since the light is detected by the light detector must satisfy γ <L _2/2 condition. Therefore, the expression (67) must be rewritten as the following expression (68).

Equation (68) also represents a condition for a maximum allowable deviation ε _{(0) 1} that indicates how far the light can be out of focus because light emitted from the light source can be detected by the photodetector. . Therefore, equation (68) can be rewritten as the following equation (69) with respect to ε _{(0) 1} .

When the dispersion (ε _{(0) 1} ) of the light emitted from the light source satisfies the condition of the equation (69), the light is detected by the photodetector. If the condition of equation (69) is not satisfied, the dispersed light circulates infinitely inside the cavity and then disappears in the optical cavity. Further, when the light from the light source is dispersed, the intensity of the light that actually contributes to the gas concentration measurement can be calculated based on the equation (69).

Theoretically, light emitted from a point light source is isotropic. However, in practice, the light source emits light according to a Gaussian pattern in which the strongest light is emitted in a specific direction and the intensity of light decreases around this direction depending on the state. According to the present invention, the point light source includes a concave mirror or lens, which not only emits the strongest light in a specific direction, but also adjusts the concave mirror or lens to generate parallel light or light that converges to a single point. Can be provided. However, the components of such a gas cell may be displaced due to external impacts or manufacturing errors. In such a case, the light emitted from the light source may travel out of the desired optical path. Therefore, the stability of the gas cell against such a light shift is calculated based on the equation (69). For example, in the gas cell shown in FIG. 1, when the values of the equation (69) are p = 15 mm, p ′ = 13.5 mm, T = −0.9, L ₂ = 4 mm, α ₀ = p-p ′, ε _{(0 ) 1} can be expressed as:

  That is, even if the light deviates from the optical path within the range of 2.29 mm in the x-axis direction, the light can be detected by the photodetector. This result only considers one direction (+ x axis), so if the -x axis is also taken into account, even if the light is dispersed within a total range of 4.58 mm, the light is stably detected by the photodetector. Will be able to.

となり、(29)式の条件を満足する。そして、(32)式を利用して、循環の回数を算出すると次のようになる。

6. Example of Analysis of Optical Cavity Manufactured Using the Present Invention An analysis method of an optical cavity manufactured according to the present invention will be described by applying equations (26), (29), and (32).
Producing optical cavities with focal lengths given as p = 15mm and p '= 13.5mm respectively, beam diameter (L ₁ ) and photodetector diameter (L ₂ ) L ₁ = 4mm and L ₂ = 4mm respectively In such a case, first, the degree of overlap between the beam diameter and the size of the photodetector is examined using equation (29). Here, T = -0.9, so

Thus, the condition of equation (29) is satisfied. Then, using the equation (32), the number of circulations is calculated as follows.

すなわち、光経路の長さは、約114cmと算出される。 Therefore, the light emitted from the light source circulates 9 times and is detected by the photodetector in the 10th circulation. In such a case, the length (L) of the optical path can be obtained using equation (26).

That is, the length of the optical path is calculated to be about 114 cm.

Table 1 presents various parameters when a gas cell is manufactured using the present invention. (In the table, L ₁ and L ₂ are 4 mm.)

  For example, if it is necessary to manufacture a 50 mm × 25 mm gas cell using the present invention, a gas cell having an optical path of about 1590 mm (= 1.59 m) can be obtained by applying the fifth condition in Table 1. Manufactured.

  What has been described above is merely an exemplary embodiment of the present invention, and the present invention is not limited to the above-described embodiment. Various modifications can be made by those skilled in the art without departing from the spirit of the invention described in the appended claims.

FIG. 5 is a diagram of an optical cavity according to one embodiment of the present invention using two two-dimensional parabolic concave mirrors. FIG. 5 is a diagram of a quadratic function parabola that forms an optical cavity according to one embodiment of the present invention. FIG. 6 is a diagram illustrating a path characteristic of an optical cavity according to an embodiment of the present invention. FIG. 5 is a diagram illustrating a method for calculating a light path condition related to a size of a light source and a light detector installed in an optical cavity according to an embodiment of the present invention. Fig. 4 shows a procedure for analyzing the optical path when light travels out of focus of an optical cavity according to an embodiment of the invention having two parabolas of the same focal length. FIG. 6 shows a procedure for analyzing an optical path when light travels out of focus of an optical cavity according to an embodiment of the present invention including two parabolas having different focal lengths. FIG.

Claims (12)

An optically closed optical cavity formed by arranging two concave mirrors so as to face each other, light incident in the optical cavity is alternately reflected by the two concave mirrors facing each other , and the two concave mirror is parabolic concave mirror, the parabolic gas cell characterized that you confocal and optical axis of the parabolic concave mirror. The parabolas of the parabolic concave mirror have different focal lengths, and by providing a light source at an arbitrary point on the concave mirror with the longer focal length, the light emitted from the light source toward the focal point can be The gas cell according to claim 1 , wherein the gas cell can converge on the optical axis after being circulated in the optical cavity while being reflected by a parabolic concave mirror. The gas cell according to claim 2 , wherein the path of light changes according to the ratio of the focal lengths of the parabolas. Comprising a photodetector for detecting light incident on the optical cavity from the light source;
The length (L) of the light path in the optical cavity until it is detected by the photodetector is given by
L = 4Np (1-T) = 4N (p + p ')
(N = number of light cycles, p, p '= focal length of each parabola, T = -p' / p)
The gas cell according to claim 2 , which is determined by: The position of the light source is A ₀ = (α ₀ , β ₀ ), and the position of the point on the concave mirror where the light that has circulated once through the optical cavity is reflected by the concave mirror is A ₁ = (α ₁ , β ₁ ), And the position of the photodetector that detects light circulating N times in the optical cavity is A _N = (α _N , β _N ), the beam diameter of the light and the cross-sectional radius of the photodetector are the following equation _{β 0 - β 1> L 1} /2 + (L 1/2) sinθ and β _N> L _2/2
(L ₁ = beam diameter of light, L ₂ = section radius of the photodetector, θ = incident angle of light from the light source with respect to the normal direction of the optical axis)
The gas cell according to claim 4 , wherein: When the position of the point at which the light from the light source is reflected first is B ′ = (− α ₀ + ε ₁ , −β ₀ + δ ₁ ), ε ₁ representing the degree of dispersion of the light from the light source The value is the following expression

The gas cell according to claim 5 , wherein: Arranging two concave mirrors facing each other to form an optically closed optical cavity;
Installing a light source in the optical cavity;
A light detector for detecting light incident on the optical cavity from the light source viewed including the steps of placing the above optical cavity,
In the optical cavity forming step, the optical cavity is formed using two parabolic concave mirrors, and the two concave mirrors are arranged so that each parabola of the parabolic concave mirror shares a focal point and an optical axis. A method for manufacturing a gas sensor. In the optical cavity forming step, the two concave mirrors are arranged so that the focal lengths of the parabolas of the parabolic concave mirrors are different from each other,
In the light source installation step, by installing a light source at an arbitrary point on the concave mirror having a longer focal length, the light emitted from the light source toward the focal point is reflected by the two parabolic concave mirrors. 8. The method of manufacturing a gas sensor according to claim 7 , wherein the gas sensor is configured to converge on the optical axis after circulating in the optical cavity. 9. The method of manufacturing a gas sensor according to claim 8 , wherein in the optical cavity forming step, the optical path is adjusted by adjusting a ratio of focal lengths of the parabolas. The length (L) of the light path in the optical cavity until it is detected by the photodetector is given by
L = 4Np (1-T) = 4N (p + p ')
(N = number of light cycles, p, p '= focal length of each parabola, T = -p' / p)
The method of manufacturing a gas sensor according to claim 8 , further comprising a step of adjusting a ratio of focal lengths of the parabolas so as to satisfy the above. In the light source installation step, the position of the light source is A ₀ = (α ₀ , β ₀ ), and the position of the point on the concave mirror at which the light circulated once through the optical cavity is reflected by the concave mirror is A ₁ = When (α ₁ , β ₁ ) and the position of the photodetector for detecting light circulating N times in the optical cavity is A _N = (α _N , β _N ), the beam diameter of the light is formula _{β 0 - β 1> L 1} /2 + (L 1/2) sinθ
(L ₁ = beam diameter of light, θ = incident angle of light from the light source with respect to the normal direction of the optical axis)
So that the light source is installed in the optical cavity,
In the photodetector installed step, wherein cross-sectional radius of the optical detector of the following β _N> L _2/2
(L ₂ = cross-sectional radius of the above photodetector)
The method of manufacturing a gas sensor according to claim 10 , wherein the photodetector is installed in the optical cavity so as to satisfy the above. When the position of the point where the light from the light source is first reflected is B ′ = (−α ₀ + ε ₁ , −β ₀ + δ ₁ ),
In the light source installation step, the value of ε ₁ representing the degree of dispersion of light from the light source is expressed by the following equation:

The gas sensor manufacturing method according to claim 11 , wherein the light source is installed in the optical cavity so as to satisfy the above.

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